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### Lesson 6-3

5-Minute Check on Lesson 6-2

- Determine whether the triangles are similar.Justify your answer.
- The quadrilaterals are similar. Write a similarity statement and find the scale factor of the larger to the smaller quadrilateral.
- The triangles are similar. Find x and y.
- 4. Which one of the following statements is always true?

Yes: corresponding angles corresponding sides have same proportion

ABCD ~ HGFE

Scale factor = 2:3

x = 8.5, y = 9.5

Standardized Test Practice:

Two rectangles are similar

A

Two right triangles are similar

B

Two acute triangles are similar

C

Two isosceles right triangles are similar

D

D

Click the mouse button or press the Space Bar to display the answers.

Similar Triangles

Objectives

- Identify similar triangles
- Use similar triangles to solve problems

Vocabulary

- None new

Theorems

- Postulate 6.1: Angle-Angle (AA) SimilarityIf two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar
- Theorem 6.1: Side-Side-Side (SSS) SimilarityIf all the measures of the corresponding sides of two triangles are proportional,then the triangles are similar

Theorems Cont

- Theorem 6.2: Side-Angle-Side (SAS) Similarity

If the measures of two sides of a triangle are proportional to the measures of two corresponding side of another triangle and the included angles are congruent, then the triangles are similar

- Theorem 6.3: Similarity of triangles is reflexive, symmetric, and transitive
- Reflexive: ∆ABC ~ ∆ABC
- Symmetric: If ∆ABC ~ ∆DEF, then ∆DEF ~ ∆ABC
- Transitive: If ∆ABC ~ ∆DEF and ∆DEF ~ ∆GHI, then ∆ABC ~ ∆GHI

P

Q

B

C

R

AA Triangle SimilarityThird angle must be congruent as well(∆ angle sum to 180°)

From Similar Triangles

Corresponding Side Scale Equal

AC AB BC

---- = ---- = ----

PQ PR RQ

If Corresponding Angles Of Two Triangles Are Congruent, Then The Triangles Are Similar

mA = mP

mB = mR

P

Q

B

C

R

SSS Triangle SimilarityFrom Similar Triangles

Corresponding Angles Congruent

A P

B R

C Q

If All Three Corresponding Sides Of Two Triangles Have Equal Ratios, Then The Triangles Are Similar

AC AB BC

---- = ---- = ----

PQ PR RQ

P

Q

B

C

R

SAS Triangle SimilarityIf The Two Corresponding Sides Of Two Triangles Have Equal Ratios And The Included Angles Of The Two Triangles Are Congruent, Then The Triangles Are Similar

AC AB

---- = ---- and A P

PQ PR

Example 1a

In the figure, AB // DC, BE = 27, DE= 45, AE = 21, and CE = 35. Determine which triangles in the figure are similar.

Since AB ‖ DC, then BAC DCE by the Alternate Interior Angles Theorem.

Vertical angles are congruent, so BAE DEC.

Answer: Therefore, by the AA Similarity Theorem, ∆ABE ∆CDE

I

Example 1bIn the figure, OW = 7, BW = 9, WT = 17.5, and WI = 22.5. Determine which triangles in the figure are similar.

Since because they are alternate interior angles. By AA Similarity, Using the definition of similar polygons,

Example 2aALGEBRA: Given RS // UT, RS=4, RQ=x+3, QT=2x+10, UT=10, find RQ and QT

Example 2a cont

Substitution

Cross products

Distributive Property

Subtract 8x and 30 from each side.

Divide each side by 2.

Now find RQ and QT.

Example 3a

INDIRECT MEASUREMENT Josh wanted to measure the height of the Sears Tower in Chicago. He used a 12-foot light pole and measured its shadow at 1 P.M. The length of the shadow was 2 feet. Then he measured the length of the Sears Tower’s shadow and it was 242 feet at that time. What is the height of the Sears Tower?

Assuming that the sun’s rays form similar ∆s, the following proportion can be written.

Example 3a cont

Now substitute the known values and let x be the height of the Sears Tower.

Substitution

Cross products

Simplify.

Answer: The Sears Tower is 1452 feet tall.

Example 3b

INDIRECT MEASUREMENTOn her trip along the East coast, Jennie stops to look at the tallest lighthouse in the U.S. located at Cape Hatteras, North Carolina. At that particular time of day, Jennie measures her shadow to be 1 feet 6 inches in length and the length of

the shadow of the lighthouse to be 53 feet 6 inches. Jennie knows that her height is 5 feet 6 inches. What is the height of the Cape Hatteras lighthouse to the nearest foot?

Answer: 196 ft

Summary & Homework

- Summary:
- AA, SSS and SAS Similarity can all be used to prove triangles similar
- Similarity of triangles is reflexive, symmetric, and transitive
- Homework:
- Day 1: pg 301-302: 6-8, 11-15
- Day 2: pg 301-305: 9, 18-21, 31

1) 2)

3) 4)

Similar Polygons

Similar Triangles (determine if similar and list in proper order)

QUIZ Prep

x

12

3

4

x + 7

-4

x - 12

6

=

=

14

10

x + 2

5

=

28

z

7

3

=

B

H

K

10

4

A

B

G

N

A

6

10

8

y + 1

12

x + 1

M

5

D

C

W

16

C

D

J

L

x - 3

P

x + 3

W

E

V

A

6

35°

S

Z

T

C

40°

1

Q

11x - 2

85°

F

B

R

S

W

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